A metallic element crystallizes into a lattice containing a sequence of layers of ABABAB ............ Any packing of spheres leaves out voids in the lattice. The percentage by volume of empty space of this is
(4) 16 %
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(1) The hexagonal base consists of six equilateral triangles, each with side 2r and altitude 2r sin 60°.
Hence, area of base =
The height of the hexagonal is twice the distance between closest packed layers.
The latter can be determined to a face centred cubic lattice with unit cell length a. In such a lattice, the distance between closest packed layers is one third of the body diagonal, i.e.
Now, in the face centred lattice, atoms touch one another along the face diagonal,
With this, the height of hexagonal becomes :
Volume of hexagonal unit is, V = (base area) × (height)
In one hexagonal unit cell, there are 6 atoms as described below :
• 3 atoms in the central layer which exclusively belong to the unit cell.
• 1 atom from the centre of the base. There are two atoms of this type and each is shared between two hexagonal unit cells.
• 2 atoms from the corners. There are 12 such atoms and each is shared amongst six hexagonal unit cells.
Now, the volume occupied by atoms =
Fraction of volume occupied by atoms
Fraction of empty space =
Percentage of empty space = 26%